P(1, 4)

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One of important consequences of the study of the non-conjugate subalgebras of the Lie algebra of the group P(1, 4) is that the Lie algebra of the group. P(1, 4) ...
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VXWZY\[ ] ^`_ [ a It is established which functional bases of the first-order differential invariants of the splitting and non-splitting subgroups of the Poincaré group P (1, 4) are invariant under the subgroups of the extended Galilei e 3) ⊂ P (1, 4). The obtained sets of functional bases are clasgroup G(1, sified according to dimensions.

bdcfe gXh\i`j0k0l0m`h\n j0g It is well known (see, for example, [10, 11, 12, 13]), that functional bases of differential invariants of Lie groups of the point transformations play an important role in group analysis of differential equations, theorethical and mathematical physics, geometry, etc. The group P (1, 4) is the group of rotations and translations of the fivedimensional Minkowski space M (1, 4). Some applications of this group in the theoretical and mathematical physics can be found in [7, 8, 9]. Continuous subgroups of the group P (1, 4) have been described in [3, 4, 6]. One of important consequences of the study of the non-conjugate subalgebras of the Lie algebra of the group P (1, 4) is that the Lie algebra of the group P (1, 4) contains, as subalgebras, the Lie algebra of the Poincaré group P (1, 3) (group symmetry of relativistic physics) and the Lie algebra of the extended e 3) (group symmetry of non-relativistic physics) (see also [7]). Galilei group G(1, Recently the functional bases of the first-order differential invariants for all continuous subgroups of the group P (1, 4) have been constructed. Some of them can be found in [2, 1]. The present paper is devoted to the classification of the functional bases of the first-order differential invariants of continuous subgroups of the group P (1, 4). It is established which functional bases of the first-order differential AMS (2000) Subject Classification: 17B05, 17B81.

o0pq0r s"rdtvuvwvx=y{zZ|v}v~Zq0r € rvtvuvwvx=y{zZ|v}v~ invariants of the splitting and non-splitting subgroups of the group P (1, 4) are e 3) ⊂ P (1, 4). invariant under the subgroups of the extended Galilei group G(1, The obtained sets of functional bases are classified according to dimensions. In order to present some of obtained results, we consider the Lie algebra of the group P (1, 4).

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P (1, 4)

‡Bg0k†n h\Ž†g0j0g0\mXj0g`\l0Š0‡‘h\„Ž0l0‹0‡B‰ Š0„0‹0i`‡BŽ

The Lie algebra of the group P (1, 4) is given by the 15 basis elements Mµν = −Mνµ (µ, ν = 0, 1, 2, 3, 4) and Pµ0 (µ = 0, 1, 2, 3, 4), satisfying the commutation relations  0  Pµ , Pν0 = 0,  0  Mµν , Pσ0 = gµσ Pν0 − gνσ Pµ0 ,  0  0 0 0 0 0 = gµρ Mνσ + gνσ Mµρ − gνρ Mµσ − gµσ Mνρ , Mµν , Mρσ where g00 = −g11 = −g22 = −g33 = −g44 = 1, gµν = 0, if µ 6= ν. Here, and in 0 what follows, Mµν = iMµν . All non-conjugate subalgebras of the Lie algebra of the group P (1, 4) are divided into splitting and non-splitting ones. Splitting subalgebras Pi,a of the Lie algebra of the group P (1, 4) can be written in the following form: ◦

Pi,a = Fi + Nia , where Fi are subalgebras of the Lie algebra of the group O(1, 4), Nia are sub◦

algebras of the Lie algebra of the translations group T (5) ⊂ P (1, 4) and + is the semi-direct sum. Non-splitting subalgebras Pej,k are subalgebras, for which a basis can be chosen in the form: X X ek = Bk + B cki Xi , drj Xj , i

j

where cki and drj are fixed real constants (not equal zero simultaneously). Bk are bases of subalgebras of the Lie algebra of the group O(1, 4), Xi are bases of subalgebras of the Lie algebra of the group T (5). We consider the following representation of the Lie algebra of the group P (1, 4): P00 =

∂ , ∂x0

P10 = −

∂ , ∂x1

0 Mµν = − (xµ Pν0 − xν Pµ0 ).

P20 = −

∂ , ∂x2

P30 = −

∂ , ∂x3

P40 = −

∂ , ∂x4

’”“A• } “ z{– — x “v˜™Xšv˜› u › x • – |vu • — y › – œ x=y{wvuvy wv— • • uvy{u “ – — ˜™ — “=d˜ y{— ˜“ – › o0ž Further, we will use the following basis elements: 0 G = M40 , 0 0 0 L1 = M32 , L2 = −M31 , L3 = M21 , 0 0 Pa = M4a − Ma0 , (a = 1, 2, 3), 0 0 Ca = M4a + Ma0 , 1 0 X0 = (P0 − P40 ), 2

(a = 1, 2, 3), Xk = Pk0

(k = 1, 2, 3),

X4 =

1 0 (P + P40 ). 2 0

e 3) is generated by the following basis eleThe Lie algebra of the group G(1, ments: L 1 , L 2 , L 3 , P 1 , P 2 , P 3 , X0 , X1 , X2 , X3 , X4 .

Ÿ0cf‚"ƒ0„ Œ\n i`ŽXh\\j0i`k0„0ik0n Œ Œ\„0i`„0gXh\n ‡B‰/n g;¡@‡Bi`n ‡BgXh\Ž¢jXŒŽ00‰ n h h\n g0Š¢Ž0l0‹0Š0i`j0l00Ž£jXŒQh\ƒ0„£Š0i`j0l0 P (1, 4)

The first-order differential invariants J of any non-conjugate k-parametrical subgroup of the group P (1, 4) can be obtained as solutions of the following systems of differential equations:  e1 J(x0 , x1 , x2 , x3 , x4 , u, u0 , u1 , u2 , u3 , u4 ) = 0,  X    e X2 J(x0 , x1 , x2 , x3 , x4 , u, u0 , u1 , u2 , u3 , u4 ) = 0, ..   .   e Xk J(x0 , x1 , x2 , x3 , x4 , u, u0 , u1 , u2 , u3 , u4 ) = 0,

e1 , X e2 , . . . , X ek , (k = 1, . . . , 12, 15)} are one times prolonged basis opwhere {X erators of any k-dimensional subalgebras of the Lie algebra of group P (1, 4), u ∂u , µ = 0, 1, 2, 3, 4. is an arbitrary smooth function on M (1, 4), uµ ≡ ∂x µ Any solution of this system can be written in the following form: J(x0 , x1 , x2 , x3 , x4 , u, u0 , u1 , u2 , u3 , u4 ) = F (J1 , J2 , . . . , Jt ),

where {J1 , J2 , . . . , Jt } is a functional basis of the first-order differential invariants of the considered subalgebra, F is an arbitrary smooth function. In this formula Ji = Ji (x0 , x1 , x2 , x3 , x4 , u, u0 , u1 , u2 , u3 , u4 ),

i = 1, . . . , t.

More details about solutions construction of the above mentioned type systems as well as the solved examples can be found in [10, 12, 13]. Using the prolongation theory (see, for example, [12, 13]) we have constructed the first prolongation for basis operators of the Lie algebra of the group P (1, 4). One times prolonged bases operators of the Lie algebra of the group P (1, 4) have the following form:

o0oq0r s"rdtvuvwvx=y{zZ|v}v~Zq0r € rvtvuvwvx=y{zZ|v}v~ e = − x 4 ∂ − x0 ∂ + u4 ∂ + u0 ∂ , G ∂x0 ∂x4 ∂u0 ∂u4

e 1 = x3 ∂ − x2 ∂ + u3 ∂ − u2 ∂ , L ∂x2 ∂x3 ∂u2 ∂u3

e 2 = − x 3 ∂ + x1 ∂ − u3 ∂ + u1 ∂ , L ∂x1 ∂x3 ∂u1 ∂u3 e 3 = x2 ∂ − x1 ∂ + u2 ∂ − u1 ∂ , L ∂x1 ∂x2 ∂u1 ∂u2

∂ ∂ ∂ ∂ ∂ ∂ Pe1 = x1 + (x0 + x4 ) − x1 − u1 − (u0 − u4 ) − u1 , ∂x0 ∂x1 ∂x4 ∂u0 ∂u1 ∂u4 ∂ ∂ ∂ ∂ ∂ ∂ Pe2 = x2 + (x0 + x4 ) − x2 − u2 − (u0 − u4 ) − u2 , ∂x0 ∂x2 ∂x4 ∂u0 ∂u2 ∂u4 ∂ ∂ ∂ ∂ ∂ ∂ + (x0 + x4 ) − x3 − u3 − (u0 − u4 ) − u3 , Pe3 = x3 ∂x0 ∂x3 ∂x4 ∂u0 ∂u3 ∂u4

e1 = − x1 ∂ − (x0 − x4 ) ∂ − x1 ∂ + u1 ∂ + (u0 + u4 ) ∂ − u1 ∂ , C ∂x0 ∂x1 ∂x4 ∂u0 ∂u1 ∂u4 ∂ ∂ ∂ ∂ ∂ ∂ e2 = − x2 C − (x0 − x4 ) − x2 + u2 + (u0 + u4 ) − u2 , ∂x0 ∂x2 ∂x4 ∂u0 ∂u2 ∂u4 e3 = − x3 ∂ − (x0 − x4 ) ∂ − x3 ∂ + u3 ∂ + (u0 + u4 ) ∂ − u3 ∂ , C ∂x0 ∂x3 ∂x4 ∂u0 ∂u3 ∂u4   e1 = − ∂ , e2 = − ∂ , e0 = 1 ∂ + ∂ , X X X 2 ∂x0 ∂x4 ∂x1 ∂x2   e3 = − ∂ , e4 = 1 ∂ − ∂ . X X ∂x3 2 ∂x0 ∂x4

e1 , X e2 , . . . , In the mentioned above denotations this basis can be written as {X e15 }. X Going through the list of all non-conjugate subalgebras of the Lie algebra of the group P (1, 4) presented in [5] one derives that the set of functional bases of the first-order differential invariants of the splitting subgroups of the group P (1, 4) contains 99 ones which are invariant under the splitting subgroups of the e 3) ⊂ P (1, 4). It is impossible to present all these extended Galilei group G(1, bases here. Therefore, below we give only a short review of the results obtained. In each example we write the basis elements of the splitting subalgebras of the e 3) and their functional basis. Lie algebra of the group G(1,

’”“A• } “ z{– — x “v˜™Xšv˜› u › x • – |vu • — y › – œ x=y{wvuvy wv— • • uvy{u “ – — ˜™ — “=d˜ y{— ˜“ – › o0¤ 1. There is 1 three-dimensional functional basis hX1 ≡ P1 , X2 ≡ P2 , X3 ≡ P3 , X4 ≡ X0 , X5 ≡ X1 , X6 ≡ X2 , X7 ≡ X3 , X8 ≡ X4 i, hX1 ≡ L3 − P3 , X2 ≡ P1 , X3 ≡ P2 , X4 ≡ X0 , X5 ≡ X1 , X6 ≡ X2 , X7 ≡ X3 , X8 ≡ X4 i, hX1 ≡ L3 , X2 ≡ P1 , X3 ≡ P2 , X4 ≡ P3 , X5 ≡ X0 , X6 ≡ X1 , X7 ≡ X2 , X8 ≡ X3 , X9 ≡ X4 i, hX1 ≡ L1 , X2 ≡ L2 , X3 ≡ L3 , X4 ≡ P1 , X5 ≡ P2 , X6 ≡ P3 , X7 ≡ X0 , X8 ≡ X1 , X9 ≡ X2 , X10 ≡ X3 , X11 ≡ X4 i, J1 = u, uµ ≡

J2 = u20 − u21 − u22 − u23 − u24 ,

∂u , ∂xµ

J3 = u 0 − u 4 ;

(µ = 0, 1, 2, 3, 4).

2. There are 4 four-dimensional functional bases. For example hX1 ≡ L3 , X2 ≡ P3 , X3 ≡ X0 , X4 ≡ X1 , X5 ≡ X2 , X6 ≡ X3 , X7 ≡ X4 i, J1 = u, J3 = u 0 − u 4 ,

J2 = u20 − u21 − u22 − u23 − u24 , J4 = u21 + u22 .

3. There are 12 five-dimensional functional bases. For example hX1 ≡ P1 , X2 ≡ P2 , X3 ≡ X1 , X4 ≡ X2 , X5 ≡ X3 , X6 ≡ X4 i, hX1 ≡ L3 , X2 ≡ P1 , X3 ≡ P2 , X4 ≡ X1 , X5 ≡ X2 , X6 ≡ X3 , X7 ≡ X4 i, J1 = u, J4 = u 3 ,

J2 = u20 − u21 − u22 − u23 − u24 , J5 = u 0 − u 4 .

J3 = x 0 + x 4 ,

4. There are 19 six-dimensional functional bases. For example hX1 ≡ L1 , X2 ≡ L2 , X3 ≡ L3 , X4 ≡ X0 , X5 ≡ X4 i, J2 = u20 − u21 − u22 − u23 − u24 ,

J1 = u, 1

J3 = (x21 + x22 + x23 ) 2 , J5 = u 0 ,

J4 = x 1 u 1 + x 2 u 2 + x 3 u 3 , J6 = u 4 .

5. There are 26 seven-dimensional functional bases. For example hX1 ≡ P3 , X2 ≡ X0 , X3 ≡ X3 , X4 ≡ X4 i,

o0¥q0r s"rdtvuvwvx=y{zZ|v}v~Zq0r € rvtvuvwvx=y{zZ|v}v~ J2 = u20 − u21 − u22 − u23 − u24 , J5 = u 1 ,

J1 = u, J4 = x 2 , J7 = u 0 − u 4 .

J3 = x 1 , J6 = u 2 ,

6. There are 20 eight-dimensional functional bases. For example hX1 ≡ L3 , X2 ≡ X0 , X3 ≡ X4 i, J1 = u, 1 2

J4 = (x21 + x22 ) , J7 = u 3 ,

J2 = u20 − u21 − u22 − u23 − u24 ,

J3 = x 3 ,

J5 = x 1 u 2 − x 2 u 1 , J8 = u 4 .

J6 = u 0 ,

7. There are 11 nine-dimensional functional bases. For example hX1 ≡ L3 − P3 , X2 ≡ X4 i, J1 = u,

J2 = u20 − u21 − u22 − u23 − u24 ,

J3 = x 0 + x 4 , x1 x3 J5 = arctan + , x2 x0 + x 4 x3 u3 J7 = + , x0 + x 4 u0 − u 4

J4 = (x21 + x22 ) 2 ,

1

J6 = x 1 u 2 − x 2 u 1 , J8 = u 0 − u 4 ,

J9 = u21 + u22 . 8. There are 6 ten-dimensional functional bases. For example hX1 ≡ P3 i, J1 = u, J3 = x 1 ,

J2 = u20 − u21 − u22 − u23 − u24 , J4 = x 2 ,

J5 = x 0 + x 4 , J7 = (x0 + x4 )u3 + (u0 − u4 )x3 , J9 = u 1 ,

J6 = (x20 − x23 − x24 ) 2 , J8 = u 0 − u 4 , J10 = u2 .

1

¦"cf‚"ƒ0„Œ\n i`ŽXh\\j0i`k0„0i†k0n Œ Œ\„0i`„0gXh\n ‡B‰§n g;¡@‡Bi`n ‡BgXh\Ž7jXŒ§h\ƒ0„7g0j0g0\Ž00‰ n h h\n g0Š7Ž0l0‹0Š0i`j0l00Ž'jXŒ§h\ƒ0„ Š0i`j0l0 P (1, 4) As in the Section 3, it is established that the set of functional bases of the first-order differential invariants of the non-splitting subgroups of the group P (1, 4) contains 158 ones which are invariant under the non-splitting subgroups e 3) ⊂ P (1, 4). It is impossible to present all of the extended Galilei group G(1, these bases here. Therefore, below we give only a short review of the results

’”“A• } “ z{– — x “v˜™Xšv˜› u › x • – |vu • — y › – œ x=y{wvuvy wv— • • uvy{u “ – — ˜™ — “=d˜ y{— ˜“ – › o0¨ obtained. In each example we write the basis elements of the non-splitting e 3) and their functional basis. subalgebras of the Lie algebra of the group G(1, 1. There is 1 three-dimensional functional basis

hX1 ≡ L3 − X0 , X2 ≡ P1 , X3 ≡ P2 , X4 ≡ P3 , X5 ≡ X1 , X6 ≡ X2 , X7 ≡ X3 , X8 ≡ X4 i, hX1 ≡ P1 , X2 ≡ P2 , X3 ≡ P3 + X0 , X4 ≡ L3 + βX0 , X5 ≡ X1 , X6 ≡ X2 , X7 ≡ X3 , X8 ≡ X4 i, β < 0, J1 = u, J2 = u0 − u4 , J3 = u20 − u21 − u22 − u23 − u24 ; ∂u uµ ≡ , (µ = 0, 1, 2, 3, 4). ∂xµ 2. There are 5 four-dimensional functional bases. For example hX1 ≡ L3 + d3 X3 , X2 ≡ P1 , X3 ≡ P2 , X4 ≡ X0 , X5 ≡ X1 , X6 ≡ X2 , X7 ≡ X4 i, d3 < 0, hX1 ≡ L3 − X0 , X2 ≡ P1 , X3 ≡ P2 , X4 ≡ X1 , X5 ≡ X2 , X6 ≡ X3 , X7 ≡ X4 i, J1 = u,

J2 = u3 ,

J3 = u 0 − u 4 ,

J4 = u20 − u21 − u22 − u24 .

3. There are 12 five-dimensional functional bases. For example hX1 ≡ P1 + βX3 , X2 ≡ P2 , X3 ≡ P3 + X0 , X4 ≡ X1 , X5 ≡ X2 , X6 ≡ X4 i, β > 0, J1 = u,

J2 = (x0 + x4 ) +

u1 , u0 − u 4 J5 = u20 − u21 − u22 − u23 − u24 . J3 = (x0 + x4 )2 − 2x3 + 2β

u3 , u0 − u 4

J4 = u 0 − u 4 ,

4. There are 34 six-dimensional functional bases. For example hX1 ≡ L3 + dX3 , X2 ≡ P3 , X3 ≡ X1 , X4 ≡ X2 , X5 ≡ X4 i, d < 0, J1 = x 0 + x 4 ,

J2 = u,

J3 = x3 + d arctan

J4 = u 0 − u 4 ,

J5 = u21 + u22 ,

J6 = u20 − u23 − u24 .

u1 x0 + x 4 + u3 , u2 u0 − u 4

5. There are 49 seven-dimensional functional bases. For example hX1 ≡ P1 + δX3 , X2 ≡ P2 + X3 , X3 ≡ X1 , X4 ≡ X4 i, δ > 0,

o0©q0r s"rdtvuvwvx=y{zZ|v}v~Zq0r € rvtvuvwvx=y{zZ|v}v~ J1 = x 0 + x 4 , J4 =

δu1 + u2 − x3 , u0 − u 4

x2 u2 + , x0 + x 4 u0 − u 4

J2 = u,

J3 =

J5 = u 3 ,

J6 = u 0 − u 4 ,

J7 = u20 − u21 − u22 − u24 . 6. There are 32 eight-dimensional functional bases. For example hX1 ≡ P3 + X0 , X2 ≡ X1 , X3 ≡ X4 i, J1 = x 2 , u3 J4 = (x0 + x4 ) + , u0 − u 4 J7 = u 0 − u 4 ,

J2 = (x0 + x4 )2 − 2x3 ,

J3 = u,

J5 = u 1 ,

J6 = u 2 ,

J8 = u20 − u23 − u24 .

7. There are 19 nine-dimensional functional bases. For example hX1 ≡ L3 − X4 , X2 ≡ X3 i, 1

J1 = x 0 + x 4 , J4 = x 1 u 2 − x 2 u 1 , J7 = u 3 ,

J2 = (x21 + x22 ) 2 , u1 J5 = arctan + x0 − x4 , u2 J8 = u 4 ,

J3 = u, J6 = u 0 , J9 = u21 + u22 .

8. There are 6 ten-dimensional functional bases. For example hX1 ≡ L3 − P3 + α0 X0 i, α0 < 0, 1

J1 = (x21 + x22 ) 2 , J3 = x 1 u 2 − x 2 u 1 , J5 = x 0 + x 4 − α 0 J7 = u, J9 = u21 + u22 ,

u3 , u0 − u 4

J2 = (x0 + x4 )2 + 2α0 x3 , x1 J4 = α0 arctan − x0 − x4 , x2 J6 = 2(x0 + x4 )3 + 6α0 x3 (x0 + x4 ) + 3α20 (x0 − x4 ), J8 = u0 − u4 , J10 = u20 − u23 − u24 .

As we see there are not the functional bases with dimensions less than 3 as well as ones with dimensions bigger than 10. It follows from using of the theorem on invariants of Lie groups of the point transformations for all none 3) ⊂ P (1, 4). More conjugate subgroups of the extended Galilei group G(1, details about this theorem can be found in [12, 13]. The results obtained can be used for the construction and investigation of classes of first–order differential equations (defined in the space M (1, 4)×R(u))

’”“A• } “ z{– — x “v˜™Xšv˜› u › x • – |vu • — y › – œ x=y{wvuvy wv— • • uvy{u “ – — ˜™ — “=d˜ y{— ˜“ – › o0ª e 3) ⊂ P (1, 4). R(u) is invariant under continuous subgroups of the group G(1, the axis of the dependent variable u.

«@¬v­$¬”®,¬”¯”°v¬”± [1] V.M. Fedorchuk, V.I. Fedorchuk, On the differential first-order invariants for the non-splitting subgroups of the generalized Poincaré group P (1, 4), Ann. Acad. Pedagog. Crac. Stud. Math. 4 (2004), 65 - 74. [2] V.M. Fedorchuk, V.I. Fedorchuk, On differential invariants of first- and second-order of the splitting subgroups of the generalized Poincaré group P (1, 4), Symmetry in nonlinear mathematical physics, Part 1, 2 (Kyiv, 2001), Pr. Inst. Mat. Nats. Akad. Nauk Ukr. Mat. Zastos., 43, Part 1, vol. 2, Nats¯ıonal. Akad. Nauk Ukraïni ¯Inst. Mat., Kiev, 2002, 140 - 144. [3] V.M. Fedorchuk, Nonsplit subalgebras of the Lie algebra of the generalized Poincaré group P (1, 4), Ukrain. Mat. Zh. 33 (1981), no. 5, 696 - 700. [4] V.M. Fedorčuk, Split subalgebras of the Lie algebra of the generalized Poincaré group P (1, 4), Ukrain. Mat. Zh. 31 (1979), no. 6, 717 - 722. [5] V.I. Fushchich, L.F. Barannik, A.F. Barannik, Podgruppovoi analiz grupp Galileya, Puankare i reduktsiya nelineinykh uravnenii, “Naukova Dumka”, Kiev, 1991. [6] W.I. Fushchich, A. F. Barannik, L. F. Barannik, V. M. Fedorchuk, Continuous subgroups of the Poincaré group P (1, 4), J. Phys. A: Math. Gen 18 (1985), no. 15, 2893 - 2899. [7] W.I. Fushchich, A.G. Nikitin, Symmetries of equations of quantum mechanics, Allerton Press Inc., New York, 1994. [8] W.I. Fushchych, Representations of the complete inhomogeneous de Sitter group and equations in the five-dimensional approach. I, Theor. Math. Phys. 4 (1970), no. 3, 890 - 907. [9] V.G. Kadyševski˘ı, A new approach to the theory of electromagnetic interactions, Fiz. Èlementar. Chastits i Atom. Yadra 11 (1980), no. 1, 5–39. [10] S. Lie, Ueber Differentialinvarianten, Math. Ann. 24 (1884), no. 4, 537 - 578. [11] S. Lie, G. Scheffers, Vorlesungen über continuierliche Gruppen mit geometrischen und anderen Anwendungen, Teubner, Leipzig, 1893. [12] P.J. Olver, Applications of Lie groups to differential equations, Graduate Texts in Mathematics, vol. 107, Springer-Verlag, New York, 1986. [13] L.V. Ovsiannikov, Group analysis of differential equations, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1982.

¤”²q0r s"rdtvuvwvx=y{zZ|v}v~Zq0r € rdtvuvwvx=y{zZ|v}v~ Institute of Mathematics Pedagogical University, Podchorążych 2 30-084, Kraków Poland; Pidstryhach Institute of Applied Problems of Mechanics and Mathematics of the National Ukrainian Academy of Sciences 3b Naukova Street Lviv, 79601 Ukraine E-mail: [email protected] Pidstryhach Institute of Applied Problems of Mechanics and Mathematics of the National Ukrainian Academy of Sciences 3b Naukova Street Lviv, 79601 Ukraine E-mail: [email protected]

Received: 20 November 2006; final version: 5 December 2007; available online: 12 May 2008.